Generating Random Vectors in (ℤ/pℤ) d via an Affine Random Process

Martin Hildebrand () and Joseph McCollum ()
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Martin Hildebrand: University at Albany, State University of New York
Joseph McCollum: Elms College

Journal of Theoretical Probability, 2008, vol. 21, issue 4, 802-811

Abstract: Abstract This paper considers some random processes of the form X n+1=T X n +B n (mod p) where B n and X n are random variables over (ℤ/pℤ) d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)2 steps suffice to make X n close to uniformly distributed where X 0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p b steps are needed for X n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X 0 is the zero vector.

Keywords: Random processes; Fourier transform; Upper bound lemma (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s10959-007-0135-5

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